Duality between Wilson loops and gluon amplitudes

TL;DR

This work investigates a deep duality between planar MHV gluon amplitudes and light-like Wilson loops in N=4 SYM, extending tests to two loops for rectangular and pentagonal Wilson loops and deriving all-order anomalous conformal Ward identities for light-like Wilson loops. By analyzing six-point configurations, it demonstrates that the BDS ansatz is not universally valid but that the Wilson-loop–amplitude duality persists, as confirmed by numerical comparisons with six-gluon amplitudes. The results reveal that broken (dual) conformal symmetry governs the finite parts, with remainder functions arising beyond four and five points, and they emphasize the interconnected roles of cusp anomalous dimensions, IR/UV structure, and integrability in this duality. Overall, the findings provide strong evidence that the duality holds at arbitrary n and loop order, offering a powerful organizing principle for scattering amplitudes in N=4 SYM and guiding future explorations of NMHV sectors and massive deformations.

Abstract

An intriguing new duality between planar MHV gluon amplitudes and light-like Wilson loops in N=4 super Yang-Mills is investigated. We extend previous checks of the duality by performing a two-loop calculation of the rectangular and pentagonal Wilson loop. Furthermore, we derive an all-order broken conformal Ward identity for the Wilson loops and analyse its consequences. Starting from six points, the Ward identity allows for an arbitrary function of conformal invariants to appear in the expression for the Wilson loop. We compute this function at six points and two loops and discuss its implications for the corresponding gluon amplitude. It is found that the duality disagrees with a conjecture for the gluon amplitudes by Bern et al. A recent calculation by Bern et al indeed shows that the latter conjecture breaks down at six gluons and at two loops. By doing a numerical comparison with their results we find that the duality between gluon amplitudes and Wilson loops is preserved. This review is based on the author's PhD thesis and includes developments until May 2008.

Figures (34)

• Figure 1: Feynman graphs contributing to $\langle{ C^{11}(x_{1}) \,{C}^{\dagger}_{11}(x_{2})}\rangle$ and $\left\langle{ \mathcal{K}(x_{1}) \mathcal{K}(x_{2})}\right\rangle$ at one loop. Solid lines denote the chiral superfield propagator (\ref{['N=1Phiprop']}), wiggly lines the gluon superfield propagator (\ref{['N=1Vprop']}).
• Figure 2: Additional Feynman graphs contributing to $\left\langle{ \mathcal{K}(x_{1}) \mathcal{K}(x_{2})}\right\rangle$ at one loop. Solid lines denote the chiral superfield propagator (\ref{['N=1Phiprop']}), wiggly lines the gluon superfield propagator (\ref{['N=1Vprop']}).
• Figure 3: The only connected Feynman graph contributing to $\left\langle{C^{11}(x_{1})\,C^{22}(x_{2})\,{C^{\dagger}}_{11}(x_{3})\,{C^{\dagger}}_{22}(x_{4})}\right\rangle$ at one loop. Solid lines denote the chiral superfield propagator (\ref{['N=1Phiprop']}). The dots represent the (chiral) superspace integrations at points $x_{5}$ and $x_{6}$.
• Figure 4: The one-loop ladder integral. Each line represents a propagator with the integration point given by a solid vertex. The reason for the names ladder and box is clearer in the momentum representation of the same integral.
• Figure 5: The two-loop ladder integral. The dashed line represents the numerator $x_{24}^2$.