[Paper Review] Resumming planar diagrams for the N=6 ABJM cusped Wilson loop in light-cone gauge

Daniele Marmiroli|arXiv (Cornell University)|Nov 20, 2012

Black Holes and Theoretical Physics68 references3 citations

TL;DR

This paper computes the cusp anomalous dimension in planar $χ=6$ ABJM theory using light-cone gauge, deriving the weak-coupling result $\Gamma_{\text{cusp}} = -\frac{\phi}{2}\lambda^2$ at second order in the 't Hooft coupling $\lambda$. At strong coupling, it employs Bethe-Salpeter equations for ladder diagrams of tree-level and one-loop gauge propagators, finding an exponential $\Gamma_{\text{cusp}} \sim \exp(\sqrt{\lambda\phi})$ that disagrees with AdS/CFT predictions in the $\phi$-dependence.

ABSTRACT

We analyse a light-like cusped Wilson loop in N=6 superconformal Chern-Simons theory at both weak and strong coupling in light-cone gauge. At the second order in the 't Hooft coupling $λ$ the correct cusp anomalous dimension $Γ_{ m cusp}=-ϕ/ 2 λ^2$ is recovered through a deformation of the contour that takes both rays of the cusp slightly off of the light-cone. The strong coupling behaviour is addressed by means of the Bethe-Salpeter equation for ladders of tree-level gauge propagators and ladders of one-loop corrected gauge propagators. It turns out that, as might be expected, the contribution of Chern-Simons tree-level propagators is insensitive of the cusp angle $ϕ$. On the other hand, corrected propagators lead to an exponential large $λ$ behaviour $Γ_{ m cusp} \sim \exp\sqrt{λϕ}$ which, though, disagrees with the AdS/CFT predictions in the power of $ϕ$.

Motivation & Objective

• To compute the cusp anomalous dimension in $\mathcal{N}=6$ ABJM theory at both weak and strong coupling using light-cone gauge quantization.
• To investigate the behavior of light-like cusped Wilson loops in Chern-Simons theory, particularly the role of gauge propagators in determining IR divergences.
• To test the consistency of perturbative results with AdS/CFT predictions, especially the $\sqrt{\lambda}$ scaling of $\Gamma_{\text{cusp}}$ at strong coupling.
• To analyze the impact of contour deformation and spurious poles in momentum space on the renormalization of the Wilson loop.

Proposed method

• Uses light-cone gauge to compute the one-loop corrected gauge propagator in three-dimensional $\mathcal{N}=6$ ABJM theory via Schwinger parameterization and contour integration.
• Applies the Mandelstam-Leibbrandt prescription to handle spurious poles in momentum space, ensuring consistent regularization.
• Constructs ladder diagrams of tree-level and one-loop gauge propagators using the Bethe-Salpeter equation to resum planar diagrams at strong coupling.
• Performs exact integration in Schwinger parameter space, extracting the small-$\epsilon$ UV divergence structure to isolate the cusp anomalous dimension.
• Derives the one-loop propagator as $G^{(2)\,ab}_{mn} = \left(\frac{2\pi}{k}\right)^2 N \delta^I_I D_{mn} \left[ -\frac{x^{-}}{[x^{+}]} + \frac{1}{2}\frac{x^{2}}{[x^{+}]^2}\log\left(-\frac{x^{2}}{(x^{T})^{2}}\right) \right] $.
• Evaluates double derivatives on the kernel to obtain the full gauge field propagator and extract the cusp anomalous dimension from the logarithmic divergence of the Wilson loop.

Experimental results

Research questions

• RQ1 Does the cusp anomalous dimension in $\mathcal{N}=6$ ABJM theory reproduce the expected $\lambda^2$ behavior at weak coupling in light-cone gauge?
• RQ2 How do Chern-Simons tree-level and one-loop gauge propagators contribute to the cusp anomalous dimension at strong coupling?
• RQ3 Does the Bethe-Salpeter resummation of ladder diagrams yield a cusp anomalous dimension consistent with AdS/CFT predictions at strong coupling?
• RQ4 What is the role of contour deformation and spurious poles in regulating the cusp anomalous dimension in light-cone gauge?
• RQ5 Why does the strong-coupling result $\Gamma_{\text{cusp}} \sim \exp(\sqrt{\lambda\phi})$ disagree with the AdS/CFT prediction $\Gamma_{\text{cusp}} \sim \sqrt{\lambda}/\pi$?

Key findings

• At second order in the 't Hooft coupling $\lambda$, the cusp anomalous dimension is recovered as $\Gamma_{\text{cusp}} = -\frac{\phi}{2}\lambda^2$ via contour deformation to avoid light-cone singularities.
• Chern-Simons tree-level gauge propagators contribute to the cusp anomalous dimension independently of the cusp angle $\phi$, indicating no angular dependence at this order.
• One-loop corrected gauge propagators lead to a strong-coupling behavior $\Gamma_{\text{cusp}} \sim \exp(\sqrt{\lambda\phi})$, which is inconsistent with the AdS/CFT prediction $\Gamma_{\text{cusp}} \sim \sqrt{\lambda}/\pi$.
• The Bethe-Salpeter equation for ladder diagrams successfully resums planar diagrams, but the resulting exponential dependence on $\phi$ contradicts known integrability and AdS/CFT results.
• The Mandelstam-Leibbrandt prescription for spurious poles in momentum space is naturally reproduced in coordinate space, ensuring UV finiteness of the final result.
• The one-loop corrected propagator is derived in closed form using Bessel functions and Schwinger parameters, with the cusp anomalous dimension extracted from the logarithmic UV divergence of the Wilson loop.