Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes

TL;DR

The paper derives all-loop anomalous conformal Ward identities for light-like Wilson loops in N=4 SYM and shows these identities fix the finite part of the n-cusp Wilson loop for n=4,5 (up to a constant) and constrain higher-n dependence to conformal invariants. It provides an explicit two-loop pentagon calculation, finding results that precisely match the two-loop five-gluon MHV amplitude, thus supporting the Wilson loop–gluon amplitude duality at weak coupling. The work also clarifies how cusp anomalies govern conformal symmetry breaking and outlines how higher-point cases may be further constrained by cross-ratios and collinear limits, potentially aided by integrability. Overall, it strengthens the case that polygonal light-like Wilson loops encode the finite parts of planar gluon amplitudes in N=4 SYM and lays groundwork for all-n analyses.

Abstract

Planar gluon amplitudes in N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n cusps (up to an additive constant) for n=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation for n=5. The result confirms the form predicted by the Ward identities and exactly matches the finite part of the two-loop five-gluon planar MHV amplitude. This constitutes another non-trivial test of the Wilson loop/gluon amplitude duality.

Figures (5)

• Figure 1: The Feynman diagram contributing to the one-loop divergence at the cusp point $x_i$. The double line depicts the integration contour $C_n$, the wiggly line the gluon propagator.
• Figure 2: Maximally non-abelian Feynman diagrams of different topologies ('webs') contributing to $\ln W(C_n)$. In the axial gauge, the vertex-type diagram (a) generates a simple pole; the self-energy type diagram (b) generates a double pole in $\epsilon$; diagram (c) with gluons attached to three and more segments is finite.
• Figure 3: The Feynman diagrams contributing to $\langle{\cal L}(x) W_n \rangle$ to the lowest order in the coupling. The double line depicts the integration contour $C_n$, the wiggly line the gluon propagator and the blob the insertion point.
• Figure 4: The Feynman diagrams contributing to $\ln {W(C_5)}$ to two loops. The double line depicts the integration contour $C_5$, the wiggly line the gluon propagator and the blob the one-loop polarization operator.
• Figure 5: The auxiliary Feynman diagrams defined in (\ref{['I-aux']}). The box depicts the fake three-gluon vertex (\ref{['J-integral']}).