A universal scaling function for giant graviton OPE coefficients

Technical Summary: A Universal Scaling Function for Giant Graviton OPE Coefficients

Problem Statement
The paper addresses a significant gap in the verification of the AdS/CFT correspondence: the lack of precision tests for Operator Product Expansion (OPE) coefficients at finite coupling. While the cusp anomalous dimension has been successfully determined exactly across all coupling regimes using integrability (due to its freedom from finite-size corrections), analogous precision tests for OPE coefficients have remained elusive. The primary obstacle is the difficulty in systematically accounting for all finite-size corrections in OPE calculations. The authors propose identifying a specific OPE coefficient that shares the "finite-size correction-free" property of the cusp anomalous dimension, thereby allowing for an exact determination at arbitrary coupling using asymptotic methods.

Methodology
The authors focus on the OPE coefficient, denoted $D_S(g)$, involving two maximal giant gravitons (represented by determinant operators) and a single-trace twist-two spinning operator $O_S$ with large Lorentz spin $S$ in planar $\mathcal{N}=4$ super-Yang-Mills theory.

1. Asymptotic Framework: The study utilizes the integrability-based method developed in prior works [31, 32], which reformulates the OPE coefficient as a worldsheet $g$-function. The exact expression involves an infinite-dimensional Fredholm determinant dependent on $Y$-functions.
2. Large Spin Limit ($S \to \infty$): The authors analyze the asymptotic expression in the limit of large spin. They demonstrate that the OPE coefficient exhibits a power-law scaling behavior:
   $$D_S(g) \sim 2^{-S + \frac{1}{2}} S^{d(g)}$$
   where $d(g)$ is the scaling function of interest.
3. Finite-Size Correction Analysis: A critical step involves proving that the exponent $d(g)$ is independent of the twist $L$ (the length of the operator). By analyzing the asymptotic Bethe ansatz equations (BAEs) and the ratio of Gaudin-like determinants, the authors show that in the large spin limit, all dependence on the twist $L$ cancels out. Specifically, the density of Bethe roots becomes universal and independent of $L$, and the ratio of determinants reduces to a series of integrals involving only the scattering kernel.
4. Computational Strategy: Assuming $d(g)$ is free from finite-size corrections, the authors compute it by taking the large spin limit of the all-loop asymptotic expression. This involves:
   • Solving the Beisert-Eden-Staudacher (BES) equation for the density of Bethe roots at arbitrary coupling.
   • Evaluating the prefactor (involving the boundary dressing phase $\sigma_B$) and the ratio of determinants in the $S \to \infty$ limit.
   • Notably, they find that higher-loop corrections to the ratio of determinants vanish in the large spin limit, simplifying the calculation significantly.

Key Contributions and Results

• Identification of a Universal Quantity: The paper provides strong evidence that the exponent $d(g)$ in the large spin scaling of the giant graviton OPE coefficient is free from finite-size corrections. This renders $d(g)$ accessible via asymptotic results, bypassing the need for full finite-size TBA calculations.
• Weak Coupling Expansion: The authors compute the perturbative expansion of $d(g)$ up to four-loop order:
  $$d(g) = \frac{1}{4} - (4 \log 2)g^2 + \frac{4}{3}(\pi^2 \log 2 - 9\zeta(3))g^4 + \dots$$
  This result matches perfectly with existing field-theoretic computations up to three loops, serving as a non-trivial check of their integrability-based method. The four-loop term is presented as a new prediction.
• Strong Coupling Predictions: In the strong coupling regime ($g \gg 1$), where no string theory prediction currently exists for this specific observable, the authors provide the first three orders of the expansion:
  $$d(g) = -(4 \log 2)g + \left( \frac{3 \log^2 2}{\pi} + 2\sqrt{2} - \frac{1}{4} \right) + \left( \frac{1}{16\sqrt{2}} + \frac{K \log 2}{4\pi^2} \right)\frac{1}{g} + O(g^{-2})$$
  where $K$ is the Catalan constant.
• Finite Coupling Interpolation: Using a numerical adaptation of the Bessel-matrix method to evaluate the boundary dressing phase, the authors compute $d(g)$ for arbitrary finite values of the 't Hooft coupling $g$. The resulting curve demonstrates a smooth interpolation between the weak and strong coupling limits.

Significance and Claims
The paper claims that $d(g)$ furnishes a "solid foundation" for an exact test of the AdS/CFT correspondence at the level of OPE coefficients. By identifying a quantity that is free from finite-size corrections, the authors enable the use of asymptotic integrability results to make all-loop predictions.

The authors emphasize that their argument for the absence of finite-size corrections constitutes "strong evidence" rather than a rigorous mathematical proof, drawing parallels to the reasoning used for the cusp anomalous dimension. They note that the simplicity of this scaling behavior is unique to maximal giant gravitons; replacing them with non-maximal gravitons or considering different operator configurations leads to more complex scaling behaviors that are not universal.

The work establishes a concrete benchmark for future strong-coupling string theory computations, specifically involving the emission of a GKP string from a D3-brane, which would be required to independently verify the strong-coupling predictions presented here.