[Paper Review] Cusp anomalous dimension in maximally supersymmetric Yang-Mills theory

Jan Kotanski|arXiv (Cornell University)|Nov 17, 2008

Black Holes and Theoretical Physics5 references3 citations

TL;DR

This paper presents a strong coupling expansion of the cusp anomalous dimension in ${\cal N}=4$ supersymmetric Yang-Mills theory using integrability-based methods, deriving analytical coefficients up to $g^{-10}$ order. It confirms agreement with string theory predictions, including the $-\frac{3\ln 2}{2\pi}$ term, and demonstrates that the weak and strong coupling expansions are both computable to high order via the BES equation and Bessel function expansions, validating the AdS/CFT correspondence.

ABSTRACT

The main features of the cusp anomalous dimension in N=4 supersymmetri c Yang-Mills theory are reviewed. Moreover, the strong coupling expansion of the cusp derived is presented.

Motivation & Objective

• To compute the cusp anomalous dimension in ${\cal N}=4$ SYM theory at strong coupling using integrability.
• To test the AdS/CFT correspondence by comparing results from ${\cal N}=4$ SYM with string theory predictions.
• To derive analytical strong coupling expansions of the cusp anomalous dimension to high order using the BES equation and Bessel function series.
• To resolve discrepancies in early string theory calculations by confirming agreement with the numerical and analytical results from the gauge theory side.

Proposed method

• Solving the BES equation numerically via Bessel function expansion and truncation to $M$ terms, transforming the integral equation into a matrix equation.
• Expanding the fluctuation density $\widehat{\sigma}_g(t)$ in an infinite series of Bessel functions $J_n(2gt)$ to enable analytical strong coupling expansion.
• Performing an asymptotic expansion in inverse powers of $g$, with $s_n(g) = g^{-1} \sum_j g^{-j} s_n^{(j)}$, to compute coefficients order by order.
• Using the relation $\Gamma_{\rm cusp}(g) = 8g^2 \widehat{\sigma}_g(0)$ to extract the cusp anomalous dimension from the solution at $t=0$.
• Applying symbolic computation to evaluate high-order coefficients, including zeta and Dirichlet beta functions, up to $g^{-10}$.
• Analyzing the Borel ambiguity of the asymptotic series to understand non-perturbative corrections, such as $g^{1/2} e^{-2\pi g}$.

Experimental results

Research questions

• RQ1 Can the cusp anomalous dimension in ${\cal N}=4$ SYM theory be computed analytically at strong coupling using integrability?
• RQ2 Do the strong coupling coefficients of the cusp anomalous dimension match predictions from string theory, particularly the $-\frac{3\ln 2}{2\pi}$ term?
• RQ3 Is the asymptotic strong coupling expansion Borel summable, and what is the nature of its non-perturbative ambiguity?
• RQ4 Can the weak and strong coupling expansions of the cusp anomalous dimension be computed to arbitrary order in ${\cal N}=4$ SYM theory?
• RQ5 Does the agreement between gauge theory and string theory results confirm the validity of the AdS/CFT correspondence at high orders?

Key findings

• The strong coupling expansion of the cusp anomalous dimension was computed analytically up to $g^{-10}$ order, with coefficients expressed in terms of zeta and Dirichlet beta functions.
• The first two coefficients, $4g$ and $-0.661907$, match the string theory prediction $4g - \frac{3\ln 2}{2\pi}$, confirming consistency with the AdS/CFT correspondence.
• The coefficient $c_1 = \frac{3\ln 2}{4\pi}$ is the leading correction, and all higher-order coefficients $c_2$ to $c_{10}$ are negative and decrease with increasing order.
• The asymptotic series is not Borel summable, with a Borel transform pole at $u = 2\pi g$, leading to non-perturbative corrections of order $g^{1/2} e^{-2\pi g}$.
• The results confirm that the cusp anomalous dimension in ${\cal N}=4$ SYM theory can be computed to high order using integrability, and the agreement with string theory validates the AdS/CFT duality.
• The numerical solution of the BES equation via matrix inversion supports the analytical results, and the method is robust up to high orders, limited only by computational power.