Probing the dynamics of stringy flux tubes with large $R$-charge

TL;DR

This work analyzes the generalized cusp anomalous dimension Γ_L at strong coupling with a large R-charge insertion, using the AdS5×S5 string dual to unify near-BPS, antiparallel, and large-L regimes. The authors derive a fully semiclassical string solution, introduce an effective R-symmetry angle Θ_eff that governs the internal opening, and identify two parameter regions (A_L and B_L) separated by a critical curve, with a non-analytic transition as L crosses L_c^θ. They compute quadratic fluctuations around the classical solution, revealing a Lamé-type structure for uncoupled modes and a WKB treatment for the coupled ones, and establish BMN-like spectra in the large-L limit. The field theory side links small-angle results to integrated defect correlators and antiparallel-limit physics to defect fusion, offering insights into Wilson-line fusion and potential three-point function behavior in the presence of heavy bulk operators. Overall, the paper provides a comprehensive, cross-regime picture of flux-tube dynamics with large R-charge and opens avenues for quantum corrections via the Quantum Spectral Curve and related defect-CFT techniques.

Abstract

We study the generalized cusp anomalous dimension, or quark-antiquark potential on the three-sphere, in the presence of a large $R$-charge $L$ and at strong coupling. Considering the insertion of a local scalar operator of charge $L$ on a cusped Wilson loop, we investigate the general solution of the dual string configuration for any value of cusp angles, paying particular attention to the different ranges of the physical parameters and obtaining a unified description that simultaneously captures the near BPS, the antiparallel lines and the large-$L$ regimes. We observe, in the limit of small separation between the lines, a non-analytic change in the behavior of the cusp anomaly at a critical value of $L$: one crosses from a region dominated by the familiar Coulomb-like singularity to a ``deconfined" situation describing the leading Lüscher corrections at large $L$ of the generalized cusp anomalous dimension. We derive the equations for small fluctuations around the classical string solution, generalizing previous analysis of the uncharged case, and we study the behavior of the spectrum near the transition region. Finally, we speculate on the field theory origin of the transition and on its implications for the fusion of Wilson lines.

Figures (7)

• Figure 1: Cusped Wilson line with opening angle $\phi$ and a $Z^L$ insertion, see equation \ref{['WL']}. The two line segments in the figure couple differently to the scalar fields, corresponding to an angle $\theta$ in an internal space. Using a conformal transformation, this configuration can be mapped to a pair of antiparallel Wilson lines on the cylinder $\mathbb{R} \times S^3$, as in the figure on the right.
• Figure 2: Critical curve, defined implicitly by $\Theta_\text{\tiny eff}(\theta, \mathcal{L})=\pi$, for the cusp internal angle $\theta$ as a function of the rescaled $R$-charge $\mathcal{L}$. The shaded region in light blue corresponds to Region $A_L$, the light gray one to Region $B_L$.
• Figure 3: The shaded region represents the range of $\Theta_\text{\tiny eff}(\theta_, \mathcal{L})$ as $\mathcal{L}$ varies, with $\Theta_\text{\tiny eff}^\text{\tiny min}(\mathcal{L})=2\,\sqrt{m_{\text{\tiny $L$}}} K(m_{\text{\tiny $L$}})$ (blue curve) and $\Theta_\text{\tiny eff}^\text{\tiny max}(\mathcal{L})= 2 K(m_{\text{\tiny $L$}})$ (brown curve). For $\mathcal{L} > \mathcal{L}_c^\text{\tiny{max}}$, $\Theta_\text{\tiny eff}(\theta, \mathcal{L})$ is always greater than $\pi$. Below this critical value, two distinct regions emerge when $\theta$ is varied at fixed $\mathcal{L}$. The red segment corresponds to Region A$_L$ in the main text, while the green segment corresponds to Region B$_L$.
• Figure 4: Range of the possible values of $m$ in the plane $(m,\Theta_{\mathrm{eff}})$, as defined by the consistency condition \ref{['thetaeff']}.
• Figure 5: Energy $\mathcal{E}$ in \ref{['E']} as a function of the geometric angle $\phi$ at fixed internal angle $\theta = \pi/3$, shown for several values of the $R$-charge $\mathcal{L}$. The first four curves correspond to $\mathcal{L}$ values below the transition, where the system exhibits the familiar Coulomb-like divergence: as $\phi \to \pi$ (antiparallel lines limit), the energy grows without bound. In contrast, the curves corresponding to larger $\mathcal{L}$ lie above the transition and remain finite as $\phi \to \pi$. Their limiting value increases with $\mathcal{L}$, asymptotically approaching $\mathcal{E} \simeq \mathcal{L}$ for large $R$-charge, consistent with the expectation \ref{['ElargeL']}. The dashed line marks the critical value $\mathcal{L}=\mathcal{L}_c^{\theta=\pi/3}$, for which the zero of the energy lies exactly at $\phi=\pi$, leaving no interval where a divergence can arise.