From correlation functions to Wilson loops

TL;DR

The paper demonstrates that in a conformal gauge theory, an n-point correlator G_n in the light-cone limit x_{i,i+1}^2 -> 0 reduces to the expectation value of a polygonal light-like Wilson loop W[C_n], linking local operator correlators to Wilson loops. It develops a dimensionally regularized framework to control the limit and identifies a universal J-factor and cusp-induced divergences that organize the leading behavior in terms of cross-ratios and conformal data. Explicit perturbative checks in ${ m N}=4$ SYM at one and two loops confirm the proposed relation, including the role of the adjoint vs fundamental representations and the emergence of the conformally invariant remainder. The results illuminate connections between correlators, Wilson loops, and scattering amplitudes, with potential extensions to strong coupling and other conformal theories via AdS/CFT and related formalisms.

Abstract

We start with an n-point correlation function in a conformal gauge theory. We show that a special limit produces a polygonal Wilson loop with $n$ sides. The limit takes the $n$ points towards the vertices of a null polygonal Wilson loop such that successive distances $x^2_{i,i+1} \to 0$. This produces a fast moving particle that generates a "frame" for the Wilson loop. We explain in detail how the limit is approached, including some subtle effects from the propagation of a fast moving particle in the full interacting theory. We perform perturbative checks by doing explicit computations in N=4 super-Yang-Mills.

Figures (10)

• Figure 1: (a) Diagrammatic representation of the correlation function; the black dots denote the points $x_i$. (b) The distances $x^2_{i,i+1}$ go to zero. (c) We are left with a Wilson loop on a polygonal contour with null sides.
• Figure 2: Feynman diagrams of different types contributing to the correlator (\ref{['freecorr']}) at tree level. The lines denote free scalar propagators $\langle{ \phi (x_i) \phi (x_j)}\rangle$. In the light-cone limit $x_{i,i+1}^2\to 0$ the leading contribution comes from diagram (a), while that of diagram (b) is suppressed by the factor $x_{34}^2 x_{1n}^2/(x_{3n}^2x_{14}^2)$.
• Figure 3: Different types of diagrams contributing to the correlation function $G_n$. Solid, wavy and dashed lines denote scalars, gluons and gluinos, respectively. The vertex with coordinate $x_i$ represents the operator ${{\cal O}}(x_i)$. The shadowed blobs stand for the rest of the diagram involving the remaining operators.
• Figure 4: The leading contribution to the correlator (\ref{['n+1co']}) in the light-cone limit. Solid, wavy and dashed lines denote complex scalars, gluons and gluinos, correspondingly. The cross denotes the insertion of the derivative of the Lagrangian with respect to the coupling constant ${L}'(x_0)$, Eq. (\ref{["L'"]}). The big blob in diagram (b) denotes the corrections to the scalar propagator shown in (c).
• Figure 5: Additional graphs with a $\phi^4-$coupling contributing to the correlation function (\ref{['kkoo']}). The contribution of the first three diagrams is given by the first line in (\ref{['Konishi']}) while the contribution of the forth diagram is given by the last line in (\ref{['Konishi']}).